967 research outputs found
On external presentations of infinite graphs
The vertices of a finite state system are usually a subset of the natural
numbers. Most algorithms relative to these systems only use this fact to select
vertices.
For infinite state systems, however, the situation is different: in
particular, for such systems having a finite description, each state of the
system is a configuration of some machine. Then most algorithmic approaches
rely on the structure of these configurations. Such characterisations are said
internal. In order to apply algorithms detecting a structural property (like
identifying connected components) one may have first to transform the system in
order to fit the description needed for the algorithm. The problem of internal
characterisation is that it hides structural properties, and each solution
becomes ad hoc relatively to the form of the configurations.
On the contrary, external characterisations avoid explicit naming of the
vertices. Such characterisation are mostly defined via graph transformations.
In this paper we present two kind of external characterisations:
deterministic graph rewriting, which in turn characterise regular graphs,
deterministic context-free languages, and rational graphs. Inverse substitution
from a generator (like the complete binary tree) provides characterisation for
prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We
illustrate how these characterisation provide an efficient tool for the
representation of infinite state systems
The Equivalence Problem for Deterministic MSO Tree Transducers is Decidable
It is decidable for deterministic MSO definable graph-to-string or
graph-to-tree transducers whether they are equivalent on a context-free set of
graphs
Ten virtues of structured graphs
This paper extends the invited talk by the first author about the virtues
of structured graphs. The motivation behind the talk and this paper relies on our
experience on the development of ADR, a formal approach for the design of styleconformant,
reconfigurable software systems. ADR is based on hierarchical graphs
with interfaces and it has been conceived in the attempt of reconciling software architectures
and process calculi by means of graphical methods. We have tried to
write an ADR agnostic paper where we raise some drawbacks of flat, unstructured
graphs for the design and analysis of software systems and we argue that hierarchical,
structured graphs can alleviate such drawbacks
MSO definable string transductions and two-way finite state transducers
String transductions that are definable in monadic second-order (mso) logic
(without the use of parameters) are exactly those realized by deterministic
two-way finite state transducers. Nondeterministic mso definable string
transductions (i.e., those definable with the use of parameters) correspond to
compositions of two nondeterministic two-way finite state transducers that have
the finite visit property. Both families of mso definable string transductions
are characterized in terms of Hennie machines, i.e., two-way finite state
transducers with the finite visit property that are allowed to rewrite their
input tape.Comment: 63 pages, LaTeX2e. Extended abstract presented at 26-th ICALP, 199
Multiple Context-Free Tree Grammars: Lexicalization and Characterization
Multiple (simple) context-free tree grammars are investigated, where "simple"
means "linear and nondeleting". Every multiple context-free tree grammar that
is finitely ambiguous can be lexicalized; i.e., it can be transformed into an
equivalent one (generating the same tree language) in which each rule of the
grammar contains a lexical symbol. Due to this transformation, the rank of the
nonterminals increases at most by 1, and the multiplicity (or fan-out) of the
grammar increases at most by the maximal rank of the lexical symbols; in
particular, the multiplicity does not increase when all lexical symbols have
rank 0. Multiple context-free tree grammars have the same tree generating power
as multi-component tree adjoining grammars (provided the latter can use a
root-marker). Moreover, every multi-component tree adjoining grammar that is
finitely ambiguous can be lexicalized. Multiple context-free tree grammars have
the same string generating power as multiple context-free (string) grammars and
polynomial time parsing algorithms. A tree language can be generated by a
multiple context-free tree grammar if and only if it is the image of a regular
tree language under a deterministic finite-copying macro tree transducer.
Multiple context-free tree grammars can be used as a synchronous translation
device.Comment: 78 pages, 13 figure
Monadic second-order definable graph orderings
We study the question of whether, for a given class of finite graphs, one can
define, for each graph of the class, a linear ordering in monadic second-order
logic, possibly with the help of monadic parameters. We consider two variants
of monadic second-order logic: one where we can only quantify over sets of
vertices and one where we can also quantify over sets of edges. For several
special cases, we present combinatorial characterisations of when such a linear
ordering is definable. In some cases, for instance for graph classes that omit
a fixed graph as a minor, the presented conditions are necessary and
sufficient; in other cases, they are only necessary. Other graph classes we
consider include complete bipartite graphs, split graphs, chordal graphs, and
cographs. We prove that orderability is decidable for the so called
HR-equational classes of graphs, which are described by equation systems and
generalize the context-free languages
The monadic second-order logic of graphs I. Recognizable sets of Finite Graphs
The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic second-order logic is recognizable, but not vice versa. The monadic second-order theory of a context-free set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins an investigation of the monadic second-order logic of graphs and of sets of graphs, using techniques from universal algebra, and the theory of formal languages. (By a graph, we mean a finite directed hyperedge-labelled hypergraph, equipped with a sequence of distinguished vertices.) A survey of this research can be found in Courcelle [ 111. An algebraic structure on the set of graphs (in the above sense) has been proposed by Bauderon and Courcelle [2,7]. The notion of a recognizable set of finite graphs follows, as an instance of the general notion of recognizability introduced by Mezei and Wright in [25]. A graph can also be considered as a logical structure of a certain type. Hence, properties of graphs can be written in first-order logic or in secondorder logic. It turns out that monadic second-order logic, where quantifications over sets of vertices and sets of edges are used, is a reasonably powerful logical language (in which many usual graph properties can be written), for which one can obtain decidability results. These decidability results do not hold for second-order logic, where quantifications over binary relations can also be used. Our main theorem states that every definable set of finite graphs (i.e., every set that is the set of finite graphs satisfying a graph property expressible in monadic second-order logic) is recognizable. * This work has been supported by the “Programme de Recherches Coordonntes: Mathematiques et Informatique.
On the Monadic Second-Order Transduction Hierarchy
We compare classes of finite relational structures via monadic second-order
transductions. More precisely, we study the preorder where we set C \subseteq K
if, and only if, there exists a transduction {\tau} such that
C\subseteq{\tau}(K). If we only consider classes of incidence structures we can
completely describe the resulting hierarchy. It is linear of order type
{\omega}+3. Each level can be characterised in terms of a suitable variant of
tree-width. Canonical representatives of the various levels are: the class of
all trees of height n, for each n \in N, of all paths, of all trees, and of all
grids
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